1. Field of the Invention
This invention relates to an approximate mode analysis method (method of approximate modal analysis) to analyze nonlinear restoring force characteristic with force-displacement hysteresis of a machine structure system. More particularly, the invention relates to the method of analysis for improving the vibration characteristic of the machine structure system by reconstructing a hysteretic characteristic using the data of a discrete load strain characteristic obtained experimentally by a vibration test or the like, alternatively, simulatively by a finite element method (FEM) analysis or the like, and by supplying an output according to the obtained data with respect to an arbitrary input.
2. Description of the Related Art
The mode analysis method has been established for its technology as a method for analyzing the vibration of the machine structure system, and has also been made commercially available as software, and frequently used. However, while the mode analysis method itself is based on a theory assuming a linear system, an actual machine structure system normally includes a plurality of nonlinear characteristics, in a bolt joined part, a sliding surface part of a baring or the like, a rotational part, a rubber pad of an engine mount as shown in FIG. 19, a joint of a robot, and so on.
With regard to a case including a nonlinear characteristic having no hysteretic characteristic, presented is a method for approximately achieving mode analysis by representing the nonlinear characteristic by an equivalent linearization method. This method has contributed to the rigidity designing and the working accuracy improvement of a machine tool, the support characteristic evaluation of a piping system and reflection in an aseismic designing method, optimal designing in support at the engine mount, and so on.
However, a restoring force characteristic typically represented by the rubber pad exhibits hysteretic nonlinearity. Conventionally, it was difficult to reduce an error when the estimated value obtained by estimating the frequency response characteristic of the machine structure system including such a nonlinear characteristic in a designing stage was compared with the frequency response characteristic obtained by an experiment.
When a vibration characteristic is requested by considering a structure of element, a characteristic for connecting elements together, and so on, in the designing stage, generally, analysis is carried out by using the finite element method. However, for a vibration characteristic, analysis is carried out mainly based on time history, and to obtain a frequency response characteristic, time history response must be repeated for each frequency step. Thus, a great deal of computing time is required even for obtaining the response of the linear system In addition, it is also possible to obtain a frequency response characteristic from the equation of motion. Regarding the case including the nonlinear characteristic, however, no general and simple methods have been presented to obtain the vibration characteristic of the entire machine structure system by presenting the nonlinear characteristic.
By carrying out the vibration experiment of the machine structure system, a frequency response characteristic including the above characteristic is obtained. In the mode analysis method, this is called an experimental mode analysis. According to this method, a frequency response characteristic is obtained by an experiment, and the system characteristic is improved by estimating an intrinsic frequency, a damping constant, a mass, spring, damping coefficients, and so on.
However, if the nonlinear characteristic is included, a frequency response characteristic is obtained in a biased manner from a linear characteristic. Consequently, because of a biased frequency, distortion of a characteristic in the vicinity of the intrinsic frequency, and so on, it is difficult to estimate an intrinsic frequency, and generally, a damping constant is excessively evaluated. As a result, wrong treatment has been taken for an improvement of the characteristic, or no proper treatment has been taken therefor.
With regard to nonlinear restoring force characteristic without hysteresis, a plural nonlinear simultaneous equation can be constructed by applying the equivalent linearization method to a multi-degree-of-freedom system equation of motion. It has been considered possible to find a solution of the equation, i.e., the frequency response characteristic of the multi-degree-of-freedom system including a plurality of nonlinear characteristics. In addition, by using a building block method (BB method) connected with the finite element method to connect the vibration characteristic of the device structure by the nonlinear restoring force characteristic, it has been considered possible to approximately carry out the mode analysis of the multi-degree-of-freedom nonlinear vibration system on the extension of the conventional mode analysis method.
However, the foregoing method was unable to deal with the system including the nonlinear restoring force characteristic with hysteresis, as typically seen in the rubber pad used when an automotive engine was mounted on a car body. In addition, the nonlinear characteristic of a bearing and the like was often represented by hysteretic nonlinear characteristic, and it was impossible to deal with this characteristic.
In the vibration characteristic analysis of the mechanical system, for the method of representing a hysteretic characteristic in a rigid part, a model for representing the hysteretic characteristic by a combination of divisional straight lines, such as bilinear model, a trilinear model or the like, a hysteretic characteristic model for representing it by a numerical expression, such as Ramberg-Osgood type, and so on, have been used. However, considering the simulation of the nonlinear characteristic of an actual machine based on the characteristics of such models, problems described below inevitably occur.
Each of the bilinear and trilinear models is a method for representing a hysteretic characteristic by the combination of straight lines. In these methods, the hysteretic characteristics to be represented are considered to be two and three inclined straight lines in sections respectively in the bilinear and trilinear models, and the shapes of the hysteretic characteristics are simulated by joining the straight lines over sections. A problem inherent in each of these methods is that since the hysteretic characteristic to be presented takes a complex shape, it cannot be sufficiently represented by the sectional combination of two to three straight lines.
In addition, since the hysteretic characteristic to be represented is considered to be two to three inclined straight lines, the inclination is changed piecewise. Consequently, even when the hysteretic characteristic to be represented takes a smooth shape, the sectional change of the inclination causes an output from the model to become unsmooth with respect to an arbitrary input.
There is also a problem inherent in the case of representing the hysteretic characteristic by the bilinear model. That is, when sectional straight line regression is made with respect to the hysteretic characteristic to be represented, depending on the position of inclination changing to be set, an obtained result varies even if a similar hysteretic characteristic is represented.
On the other hand, the hysteretic characteristic model represented by the Ramberg-Osgood type is a method for representing a hysteretic characteristic by a polynomial. This method is designed to simulate the shape of the hysteretic characteristic by deciding maximum and minimum coordinates based on a skeleton curve, and by connecting upward and downward curves from the coordinates. To represent the hysteretic characteristic by this method, it is necessary to match the hysteretic characteristic to be represented with the hysteretic characteristic model by using a least square method or-the like. However, when the hysteretic characteristic obtained from the vibration test, FEM analysis, or the like has a complex shape, it is difficult to set a parameter to match the characteristic in detail with the curve-matched hysteretic characteristic model.
Moreover, in the system represented by the hysteretic characteristic model of the Ramberg-Osgood type, for performing approximate vibration characteristic analysis by the equivalent linearization method, it is necessary to calculate Fourier series of an output with respect to an arbitrary input, as an approximation of a nonlinear factor. However, depending on a parameter for representing this model, it may be difficult to analytically calculate Fourier series.
A difference in equivalent rigidity and equivalent damping depending on the presence of a hysteretic characteristic is that while there is neither phase delay nor advancement in a relation between an input and an output in the case of absence of hysteretic characteristics, in the case of the presence of a hysteretic characteristic, a phase relation is included in a relation between an input and an output, and this need be considered when united with the equation of motion.
With regard to the representation of nonlinear hysteretic characteristic by a describing function (equivalent linearization method in the field of automatic control), in the field of an electric system, when the characteristics of a current and a magnetic field were taken into consideration, it was considered possible to numerically represent the characteristics by the describing function based on actual characteristics. However, for the application to the multi-degree-of-freedom system and the representation of the frequency response characteristic as mode analysis as seen in the machine structure system, no particular solution of a plural nonlinear simultaneous equation having a phase characteristic from the beginning has been presented, except for an area targeted from a mathematical interest.
For performing vibration characteristic analysis of the nonlinear system having a displacement strain characteristic dependent on a past history, it was necessary to apply the straight line regression, the least square method, or the like to the foregoing hysteretic characteristic model. Thus, it was difficult to match, in detail, the shape of the hysteretic characteristic model used for analysis with the data of the hysteretic characteristic obtained from the vibration test, the FEM analysis, or the like to be applied, and this point was a problem when vibration characteristic analysis was carried out.
The present invention was made to advantageously solve the foregoing problems. In accordance with the invention, a method of analysis is provided for the representation of the vibration characteristic of a machine structure system having nonlinear restoring force characteristic with hysteresis by a frequency response characteristic, the method of analysis making approximate mode analysis enable to be performed by representing the vibration characteristic numerically by equivalent rigidity and equivalent damping based on the nonlinear characteristic of an actual system, and by uniting such with a multi-degree-of-freedom equation of motion and obtaining a solution of the equation. This method contributes to the evaluation and improvement of the vibration characteristic of the machine structure system. As a result, it is possible not only to predict the characteristic of the system in a designing stage, but also to estimate a vibration system parameter more accurately and rationally from the vibration characteristic obtained by the vibration experiment of the actual system.
According to the invention, without using the method of representing a hysteretic characteristic based on the application of hysteretic characteristic data obtained from a vibration test or the like to a certain model, an output is made according to the obtained hysteretic characteristic with respect to an arbitrary input, based on the obtained hysteretic characteristic data. For such a purpose, Preisach model designed for recording a magnetic characteristic in an electric and electronic field is used to make a hysteretic characteristic model regarding the load strain of the mechanical system. Moreover, a mechanical vibration characteristic analysis including a test characteristic is performed by using this hysteretic characteristic representing method.
When a hysteretic characteristic model obtained from the vibration test or the like is made by the Preisach model, since this model includes many relay elements, it is necessary to measure a minor loop constituting this hysteretic characteristic model. However, the measurement of the minor loop of the hysteretic characteristic by the vibration test may be difficult. In addition, there is a problem when the hysteretic characteristic model obtained by the invention is used for vibration characteristic analysis. That is, when an equivalent linearization method is used because the hysteretic characteristic is numerically represented, it is impossible to analytically obtain equivalent rigidity obtained by Fourier transformation. Thus, according to the invention, equivalent rigidity is numerically obtained, and used for analysis.
According to the invention, when a solution is found to the nonlinear equation of motion of the system by Newton-Raphson method, data is obtained on the equivalent rigidity of a nonlinear factor, and this data is interpolated by a piecewise polynomial such as a spline function or the like. The spline function represented by the piecewise polynomial can be differentiated. Thus, Jacobian matrix is formed when Newton-Raphson method is applied. This process develops an output to the hysteretic characteristic in Fourier series by using a solution obtained for each repetition, making it possible to achieve high efficiency for the method of obtaining equivalent rigidity and equivalent damping.
According to the invention, the vibration characteristic analysis including the hysteretic characteristic data obtained from the vibration test, FEM analysis, or the like, can be performed. Thus, it is possible to perform analysis more closely related to an experiment. Moreover, according to the hysteretic characteristic analysis method of the invention, even a nonlinear characteristic having no history can be analyzed from data obtained from the vibration test, the FEM analysis, or the like by a similar method.
The foregoing arrangements enable a frequency response characteristic to be clarified by taking, into the equation of motion, a nonlinear characteristic existing in a connected part between element structures in a machine tool or the like, a nonlinear characteristic provided in a bearing, a sliding surface part or the like, a nonlinear characteristic of a support component used for engine suspending, a nonlinear characteristic generated in a robot joint, and so on. Accordingly, the improvement of the characteristic and dealing with the characteristic based on the above considerations can be rationally pursued. Compared with the approximate mode analysis based on a consideration given to the nonlinear characteristic having no history, more accurate estimation can be made quickly by numerically representing the hysteretic characteristic by equivalent rigidity and equivalent damping and considering the same based on an actual characteristic.
Therefore, in an age where competition is severe in an automobile industry or the like, among nations, and between corporate groups, by solving the existing problems, it is possible to realize the prompt starting of production, and to quickly deal with problems which arise.